1 Chapter 1 Promises and Pitfalls of Polarimetry 1.1 Introduction Imagine an observational technique, differential in nature, that takes full advantage of the informa- tion content a photon has to offer. Photometric conditions would be unecessary, allowing ground- based telescopes to outsrip their space-based counterparts for uses where imaging is not required. Indeed, such a technique has been around for decades in the form of polarimetry. Why, then, are the numbers of polarimeters and polarimetrists so few? Does the bright side of polarimetry simply fall on blind eyes? In 1852, Sir George Gabriel Stokes invented a formalism for decomposing the electric field oscil- lations of light that is still used today. Consider a right-handed, Cartesian coordinate system with light propagating in thez ^ direction. The electric field of this light beam varies in time as E~ (t) = Ex cos(!t − δx)^x + Ey cos(!t − δy)^y (1.1) with amplitudes and phases Ei and δi. The path of the electric field vector, when projected onto the xy plane, describes an ellipse. Such light is therefore \elliptically polarized". The Stokes parameters I; Q; U, and V are defined by the time-averaged quantities 2 2 2 I = Ex + Ey (1.2a) 2 2 Q = Ex − Ey (1.2b) U = 2 hExEy cos(δx − δy)i (1.2c) V = 2 hExEy sin(δx − δy)i : (1.2d) Thus, the Stokes I parameter describes the total intensity of the beam. Stokes Q and U are mea- sures of the \linear" polarization of the light beam, where Stokes Q represents the net electric field component along thex ^ (+Q) ory ^ (−Q) direction and Stokes ±U describes the net electric field component at ±45◦ from thex ^ direction. Stokes V , a measure of the \circular" polarization of the light beam, represents the net electric field component that rotates clockwise (+V ) or counterclock- wise (−V ) at constant angular frequency. The orthogonal basis vectors of linear polarization, Q and U, are only separated by 45◦ in physical space. Rotation of a light beam by ±90◦ reverses the sign of Q and U. Therefore, periodicity in linear polarization occurs by rotation through 180◦. When projected onto the sky, Stokes +Q points north/south, −Q east/west, +U northeast/southwest, and −U is northwest/southeast (Figure 1.1). Stokes parameters are usually normalized to the intensity of light, I. The fractional degree and position angle of net polarization are then s Q2 U 2 V 2 P ≡ + + (1.3a) I I I 1 U=I Θ ≡ arctan : (1.3b) 2 Q=I Polarimetry is therefore a differential technique, where the fractional degree of polarized light and its orientation are the relevant quantities. This is in contrast to absolute techniques such as pho- tometry, which require stringent calibration to determine whether fluctuations in data are intrinsic to the source or are due to systematic effects. To utilize photometry as a differential technique, one must monitor photometric standard stars 3 Figure 1.1: Stokes parameters projected onto the sky. The ellipse indicates a general, elliptically polarized light beam with Stokes parameters −Q, +U, and +V , where Q > U. simultaneously with the target. The non-uniformity of the Earth's atmosphere forces one to choose standard stars at a small angular distance from the target. Thus, if a pocket of turbulence passes through the line of sight of both stars roughly simultaneously, the scintillation event should be sub- tracted out. Even for high quality calibration, however, photometric precision better than one part in 103 is extremely difficult to achieve from the ground. Space-based telescopes overcome scintilla- tion from the atmosphere, but their smaller apertures ensure that even photon shot noise-limited operation rarely produces precision less than one part in 104. However, I will show in subsequent chapters that I have achieved polarimetric precision on bright stars of order one part per million. Unfortunately, polarimetry still requires calibration. The largest source of uncertainty in polari- metric measurements is usually polarization intrinsic to the telescope and instrument. Polarization of light is sensitive to the geometry of scattering as well as the optical properties of the scatterer. Therefore, asymmetries in mirror coating, as well as asymmetry in the angle of reflection integrated over the mirror surface, will generate intrinsic polarization. Analogous to dark subtraction in pho- tometry, subtraction of this telescope/instrument polarization is required. Generally this progresses by observing \unpolarized" standard stars. Since polarization is sensitive to asymmetry in the 4 source, nothing in the Universe is truly unpolarized. However, it is possible to measure polarization consistent with zero for some stars. Sky subtraction proceeds identically in polarimetry as it does in photometry, and conventional flat-fielding is required in imaging polarimetry. In addition, polarized standard stars are observed to ensure that the gain of the system is calibrated. This is effectively flat-fielding for single-pixel detectors. Since appropriate calibration can indeed be performed, what are the benefits of observing polarized light from the sky? 1.2 Promises of Polarimetry 1.2.1 Extrasolar Planets Extrasolar planets are one of the most exciting objects in astronomy to study. Questions such as \How did we get here?" and \Are we alone?" are directly applicable to the study of extrasolar planets. Regarding the former question, planet formation is the result of accretion of material in circumstellar disks. Polarimetry can provide valuable clues to the nuances of this process and will be discussed later. As for the latter question, the existence of planets around other stars has been sought since recorded history. Evidence of Earth-like, or at least life-supporting, planets could have enormous impact on virtually all aspects of society, not the least of which would be the impact on planning and funding future astronomical investigations. The first extrasolar planets were discovered around a pulsar in 1992 by observing periodic Doppler shifts in its pulses (Wolszczan & Frail 1992). These three nearly Earth-mass planets have masses 0:020 ± 0:002, 4:3 ± 0:2, and 3:9 ± 0:2 M⊕ and orbit PSR B1257+12 with periods of ≈ 25, 67, and 98 days, respectively (Konacki & Wolszczan 2003). Beginning in 1995, hundreds of close-in, Jupiter-mass planets have been detected by periodicities in stellar radial velocity (Mayor & Queloz 1995). Recently, extrasolar planet research has progressed from planet detection to the beginning stages of planet characterization. Infrared planetary emission has been directly detected (Deming et al. 2005), and dayside/nightside contrast in that emission has been observed (Knutson et al. 2007). Moreover, while the initial detected population of extrasolar planets was of order one Jupiter mass, refinement of the radial velocity technique has permitted Neptune-mass planets to be discovered (Lovis et al. 2006). 5 However, to truly begin to characterize individual planets, their most basic characteristic, mass, must be accurately determined. Since the radial velocity technique is insensitive to stellar reflex motion in the plane of the sky, estimation of precise masses for the large majority of known planets is hampered by the inability to measure orbital inclination, i. Measured mass, m, is only a lower limit to the true mass, M, because m = M sin i. Planets in edge-on orbits transit the disk of their host star every orbit, which causes a periodic dip in the stellar flux as the planet transits the disk of its parent star. The shape of the system lightcurve is indicative of orbital inclination, so inclina- tion estimates from transit observations can be coupled with radial velocity data to derive accurate masses. Indeed, masses of transiting planets can be measured with a precision of less than one Jupiter mass. The transit of HD 209458 was discovered by Charbonneau et al. (2000); since then, dozens of transiting planets have been discovered. However, the probability of transit occurrence in a sample of systems with randomly distributed inclinations scales as R∗=a, where R∗ is the stellar radius and a is the planetary semimajor axis. This is because the solid angle subtended by the transit shadow is 2π × 2R∗=a out of a total 4π steradians. Thus, transiting planets only comprise ≈ 10% of known extrasolar planets. Astrometry holds promise for determining masses of planets, because the star's motion in the plane of the sky is observed. The astrometric motion a∗ of an extrasolar planet host star is simply the star's lever arm with respect to the center of mass of the system: Mp a∗ = a : (1.4) M∗ Since typical mass ratios for extrasolar planets/host stars are of order one part in 10−3, the astro- metric motion of a star 100 parsecs away with a planet at a = 0:05 AU is of order 0.5 µas. While space-based interferometers have the potential to graze this regime, astrometric mass measurements are more likely for planets at larger semimajor axes. The same selection effect occurs for direct imaging of planetary emission, because a star's diffracted halo decreases in brightness with increase in angular distance. Therefore, orbits of extrasolar planets seen astrometrically or by direct imaging are more likely for planets at large semimajor axes. This differs from the radial velocity technique, because close-in extrasolar planets are preferen- tially selected for because of two reasons. First, stellar velocities scale as 6 2πa 2π v = = p (1.5) T a for circular orbits, where a is orbital semimajor axis. Second, close-in planets undergo more orbits in a given amount of time than do planets at larger semimajor axes, so confirmation of statistically significant periodicity requires a shorter temporal baseline. We are developing an observational technique that has the potential to determine system incli- nation for close-in extrasolar planets (so-called \hot Jupiters").